Data-Driven Strategies for selective data transmission in sensor
networks
Giorgio Battistelli, Alessio Benavoli, and Luigi Chisci
Abstract— Energy efficiency is a crucial issue for any task involving wireless sensor networks. The present paper addresses nonlinear state estimation over a centralized sensor network, i.e. a set of sensor nodes communicating with a central information fusion unit, and proposes smart data-driven strategies by which sensors decide which data transmit to the central unit so as to reduce data communication, and thus avoid congestion prob-lems as well as prolong the network lifetime, while providing enhanced performance with respect to periodic transmission. Both measurement and estimate transmission strategies are developed. To cope with nonlinear sensors that cannot fully ob-serve the state, suitable nonlinear observability decompositions are employed. A bearing-only tracking simulation case-study is presented in order to demonstrate the effectiveness of the proposed approach.
I. INTRODUCTION
The recent and rapid advances in WSN (Wireless Sensor Network) technology pose challenging estimation issues re-lated to the possibility of optimally exploiting the distributed information provided by the WSN while preserving as much as possible the limited energy resources of the wireless sensor nodes and, thus, prolonging the network lifetime. Since, as well known, data communication represents by far, for a sensor node, the most energy-consuming task, the idea of controlling data transmission so as to achieve a trade-off between communication costs and estimation performance has recently received a certain attention in the literature (see [1], [2], [3], [4], [5] and the references therein). This can be done either in a centralized or in a distributed way. In the former setting, the fusion node selects only a subset of the available sensors to receive the information [6], [7]. Conversely, in a distributed setting, each sensor node decides whether or not the data should be transmitted only on the grounds of the locally available information.
Recent work [5] has concerned a network architecture, in which the sensor nodes - equipped with processing capabil-ities - transmit data (either raw measurements or computed estimates) to the central fusion node, providing estimation strategies that properly balance data processing in the WSN nodes and data communication from the sensor nodes to the central unit. The idea in [5] is to control transmission in sensor nodes by selectively transmitting data (measurements or estimates) whose distance, in a suitably weighted norm, Alessio Benavoli acknowledges financial support from the Swiss NSF grant n. 200020-137680/1.
G. Battistelli and L. Chisci are with Dipartimento di Sistemi e Infor-matica, DSI-Universit`a di Firenze, Via Santa Marta 3, 50139 Firenze, Italy
{battistelli, chisci}@dsi.unifi.it
A. Benavoli is with Istituto “Dalle Molle” di Studi sull’Intelligenza Artificiale, 6928 Manno-Lugano, [email protected]
from a properly defined prediction computed on the basis of information available to both the sensor node and the fusion unit exceeds a given threshold chosen according to the desired transmission rate. The data-driven selective transmission strategies proposed in [5] provide significant performance improvements with respect to periodic transmis-sion operating at the same rate but are unable to cope with the presence of nonlinear sensors that cannot fully observe the target process (e.g. angle-only or range-only or Doppler-only position sensors employed for target localization/tracking). This paper extends the data-driven measurement and estimate transmission strategies to general nonlinear systems making use of nonlinear observability decompositions at the sensor level.
A. Problem Formulation
The present paper addresses estimation of the state of a discrete-time dynamical system
xk+1= f (xk) + wk (1)
given measurements collected from multiple sensors
yki = hi(xk) + vik, i = 1, . . . , s (2) under a limitation on the communication rate from each remote sensor unit to a central information fusion unit. Each remote sensor collects noisy measurements of the given system, can process them to find filtered estimates and transmits, at a reduced rate, either measurements or estimates to the fusion node. The fusion node, on the basis of the data received from the remote sensors, should provide, in the best possible way, an estimate of the system state.
In the foregoing, we formalize the concept of
communi-cation strategy (CS) with fixed rateαi for sensori. To this end, let us introduce for each sensor i binary variables ci
k such thatci
k = 1 if sensor i transmits at time k or cik = 0 otherwise. Then, a decision mechanism with rateαi∈ (0, 1) can be formally defined as any, deterministic or stochastic, mechanism of generatingci k such that lim t→∞ 1 t t X k=1 Eci k = αi (3)
where E{·} denotes the expectation operator. This indicates
that, for each sensor, the averaged number of data trans-missions per time unit is constrained to take a value αi. Such a constraint can be used to model all those practical situations in which the sensing units and the monitoring unit are remotely dislocated with respect to each other and the communication rate between them is severely limited.
Notations:: Z+ is the set of positive integers; given a square matrix M , tr(M ) denotes its trace; given a positive
definite matrix M , EM △
= {ζ : ζ′M ζ ≤ 1} denotes the ellipsoid centered at the origin associated withM ; kζkM
△
= (ζ′M ζ)1/2
is the weighted norm of the vectorζ; given
ma-trices M1, . . . , Ms, diag(M1, . . . , Ms) denotes the block-diagonal matrix whose block-diagonal blocks are M1, . . . , Ms; while col(M1, . . . , Ms) is the matrix obtained by stacking
M1, . . . , Ms one on top of the other.
II. DATA-DRIVEN STRATEGIES FOR MEASUREMENT TRANSMISSION
In the spirit of the results of [5], it is supposed that each sensor i transmits its local measurement yi
k on the basis of a data-driven strategy of the type
ci k = 0 , ifkyi k− ˜yikk2Wi k ≤ δi 1 , otherwise (4)
where the vectors y˜i
k, k ∈ Z+, the positive definite weight matrices Wi
k, k ∈ Z+, and the positive reals δi have to be chosen so as to ensure that the transmission rate constraint (3) is satisfied.
Hereafter, the process noise wk and the measurement noises vik, i = 1, . . . , s in (1)-(2) are supposed to be zero-mean stochastic processes with E{wkw′t} = Q δkt,
E{vikw′t} = 0, and E{vki(v j
t)′} = Riδijδkt where δij denotes the Kronecker delta.
A. Operations at fusion node
One of the advantages of a data-driven strategy over non data driven strategies (e.g., periodic) is that, even when no transmission is performed from the sensor node, the fusion node knows that the measurementyi
kbelongs to the ellipsoid
δiEi
Wk. Such information can be fruitfully exploited by
suit-ably modifying the nonlinear filtering algorithm implemented in the fusion node. In fact, it has been shown in [5] that the case of no transmission from sensori at time k can be treated
as if a virtual measurement zik = ˜yki were generated by the measurement channel
zki = hi(xk) + vki − uik (5) whereui
k is uniformly distributed in the ellipsoidδiEWi k and
uncorrelated with vi
k. Note that thevik− uik has zero-mean and covariance matrix1
E{(vki − uik)(vki − uik)′} = E{vik(vki)′} + E{uik(uik)′}
= Ri+ δi mi+ 2 (Wi k)−1, where mi △ = dim(yi
k). As discussed in [5], this is possible from a Bayesian point of view in that, regardless of the choice of y˜i
k, the posterior PDF of xk conditioned to the fact that no data have been received (i.e.,cik= 0) coincides with the posterior PDF of xk conditioned to the fact that
1Recall that a uniform random variable taking value in an ellipsoid E
M⊂ Rmhas covariance matrix equal to[(m + 2)M ]−1.
a measurement zi
k = ˜yki has been originated from a virtual measurement channel (5).
In view of the foregoing considerations, by defining
zi k △ = ci kyki + (1 − cik)˜yik Rki △ = Ri+ (1 − cik) δi mi+ 2 (Wki)−1 (6) and, accordingly, zk △ = col z1 k, . . . , zks , Rk △ = diag R1k, . . . , Rsk , h(·) =△ col h1(·), . . . , hs(·) ,
the estimate xˆk|k at the fusion node can be computed by means of the following recursion
ˆ
xk|k, Pk|k = upd ˆxk|k−1, Pk|k−1, zk, Rk, h ,
ˆ
xk+1|k, Pk+1|k = pred ˆxk|k, Pk|k, Q, f . (7) Here and in the following,upd(·) denotes the updating step
of some nonlinear filtering algorithm; its first and second arguments are the predicted estimate and covariance, its third argument is the measurement vector, its fourth argument is the covariance of the additive measurement noise, and its last argument is the (non)linear measurement function. Similarly,pred(·) denotes the prediction step of the nonlinear
filtering algorithm; its first two arguments are the updated estimate and covariance, its third argument is the covariance of the additive process noise, and the last argument is the (non)linear function which relates the state at time k with
the state at timek + 1.
Notice that the filtering steps pred(·) and upd(·) can
be performed via any nonlinear filter such as for instance the Extended Kalman filter, the Unscented Kalman filter (UKF), or a particle filter [8], [9]. UKF represents a good tradeoff between accuracy and computational cost and, for this reason, will be adopted in the subsequent developments.
B. Operations at sensor node
As should be evident, the performance of transmission strategy (4) may be significantly affected by the specific mechanism used for generating the quantities y˜i
k and Wki. In this connection, in view of the discussion of Section 2.1 of [5], a sensible choice can be obtained by tailoring the vectory˜i
k and the weight matrixWki to the prior distribution of the measurementyi
kso thaty˜ikcoincides with the predicted value ofyi
k andWki is proportional to the inverse innovation covariance. In this way, supposing that the PDF of yi
k is approximately characterized by a radial symmetry, the volume of the non-transmission region is close to being minimal (see Proposition 1 of [5]). Further, since the vector
˜ yi
k is used in the fusion node in order to define the vectorzk and compute the estimatexˆk|k, the mechanism for generating the vectors y˜i
k and the weight matrices Wki must use only information available to both the sensor i and the fusion
is that the volume of the non-transmission region does not grow unbounded with time. In fact, in a linear setting, it has been shown that this is a sufficient condition for preserving the stability of the estimation error at the fusion node for any positive transmission rate [5]. Summing up, taking into account such desired properties, a reasonable choice is
˜ yik= ˆyk|k−1i , (Wki)−1 = 1 tr(Si k) Ski (8)
whereyˆik|k−1is a prediction ofyik based on the information shared by the fusion node and sensori up to time t − 1 and Si
k is the corresponding covariance. The normalization factor
tr(Si
k) is a way for ensuring that the volume of the non-transmission region is always bounded in that tr(S1i
k)
Si k ≤ I. Of course other choices for the normalization factor can be devised (e.g., the spectral radius of Si
k). The main reason for preferring the trace is that it can be computed with little computational effort from the remote sensors.
The main difficulty in computing the predictionyˆi k|k−1 is that in general, even assuming collective observability from the whole set of sensors, the state vector need not be observ-able from a single sensor. This means that, locally in a remote sensor, it may not be possible to construct a stable filter for estimating the whole state of system (1). Nevertheless, it may still be possible to compute the predictionyˆi
k|k−1by resorting to an observability decomposition of the state space.
To this end, consider the noise-free system
xk+1 = f (xk) (9)
yi
k = hi(xk) (10)
and consider the set of functions
Θ△= {hi(x), hi◦ fN(x), N = 1, 2 . . .}.
wherefN denotes theN -th composition of f, i.e. the function f composed with itself N times. As well known, under
suitable assumptions, observability of system (9) from the output yi
k corresponds to the fact that dim (span dΘ) = n almost everywhere, where dΘ is the differential of Θ and n is the system order. If, instead, span dΘ has constant
dimension ni,o< n in the set of interest then observability does not hold, but it might be possible to perform a state space decomposition with respect to the system observability properties. Hereafter we shall assume that, for all the sensors for which observability does not hold, such a decomposition exists or, more precisely, that there exists a change of coordinatesq = Ti(x) such that system (9) can be rewritten as
qk+1i,o = f˜i,o(qki,o) (11)
qk+1i,¯o = f˜i,¯o(qi,o k , q i,¯o k ) (12) yik = h˜i(q i,o k ) (13)
where qi,o has dimensionni,o,qi,¯ohas dimension n − ni,o,
q = col qi,o, qi,¯o = col Ti,o(x), Ti,¯o(x), and the subsys-tem made up by (11) and (13) is observable by construction. Notice that, in general, the observability decomposition will
be different for different sensors, hence the dependence on the index i. Discussions on the existence of such a
decomposition in a discrete-time setting can be found, for instance, in [10], [11]. As evident from (11)-(13), in the noise-free case, the outputyi
kdepends only on the observable part of the system. Thus one can exploit this fact and construct a filter for the observable subsystem in order to obtain a predictionyˆi
k|k−1 of the measurement yik.
In practice, turning back our attention to the noisy case (9), each sensor will consider only the subsystem
qi,ok+1 = f˜i,o(qi,o k ) + gi(q i,o k , q i,¯o k , wk) (14) yki = ˜hi(q i,o k ) + v i k (15)
where the function g(·) has the property that g(·, ·, 0) = 0.
While, in general, in the presence of noises there is not a clear decoupling between observable and nonobservable states, several techniques can all the same be devised for constructing a filter for (14)-(15) whenever qi,¯ko belongs to a (known) compact set, which is usually the case when dealing with state estimation for physical systems. For ex-ample, the simplest alternative consists in treating the term
gi(qi,o k , q
i,¯o
k , wk) as an additive noise vector. Alternatively, a more accurate approach amounts to considering the vector
qki,¯oas a slowly time-varying unknown parameter and resort-ing to robustness arguments. Notice that when the functions
f (·) and hi(·) are smooth then the function gi(·) will be smooth as well, thus the contribution of the second term in the right hand side of(14) will in general be small for small
disturbanceswk.
Taking into account subsystem (14)-(15), the operations for computing the predictionyˆi
k|k−1in nodei are as follows
ˆ qk|ki,o, Pi k|k = upd(ˆqk|k−1i,o , Pi k|k−1, zik, Rik, ˜hi) ˆ qk+1|ki,o , Pi k+1|k = pred(ˆqk|ki,o, Pi k|k, Q, ˜fo,i, gi) ˆ yi k+1|k, Ski = proj(ˆqk+1|ki,o , Pi k+1|k, ˜hi) (16)
Hereproj(·) denotes the function that projects the predicted
estimate and relative covariance on the measurement space; its first two arguments are the predicted estimate and covari-ance and its last argument is the (non)linear function that projects the observable state in the measurement space. Such a projection can, for example, be computed by means of the unscented transform [8]. Notice that, with a little abuse of notation, we have augmented the arguments of pred(·) by
including also the (non)linear function gi. Notice also that the proposed filter does not use the true measurement yi
k and covarianceR but, instead, uses the virtual measurement zi
k and its covariance Rik as defined in (6). This choice is motivated by the fact that, as discussed in the previous section, the filter (16) should be also runned in the fusion node in order to compute the virtual measurement zk and, consequently, the fused estimatexˆk|k.
C. An example: bearing only tracking
In this section, the case of bearing only tracking is analyzed in order to show how the foregoing discussion
applies to a context of practical interest. The problem is that of estimating the state of an object whose motion is described by the kinematic nearly-constant velocity model
xk+1= 1 0 ∆ 0 0 1 0 ∆ 0 0 1 0 0 0 0 1 xk + wk,
where ∆ is the sampling interval; the unknown state vector
is given by the position and velocity components along the coordinate axes (ξ, η), i.e., x = col(ξ, η, ˙ξ, ˙η). The state has
to be estimated given measurements collected from a network of direction-of-arrival (DOA) sensors
yki = atan2 ηk− ηi, ξk− ξi + vki, i = 1, . . . , s (17) whereatan2(·) denotes the four-quadrant inverse tangent and (ξi, ηi) is the position of the i-th sensor.
As well known, it is not possible to observe the whole state from a single angular sensor. To see this, note that the
N -th composition of f (·) takes the form
fN(x) = col (ξ + N ∆vx, η + N ∆vy, vx, vy) and, consequently,
d hi◦ fN = 1 (ξN,i)2+ (ηN,i)2
×−ηN,idξ + ξN,idη − N ∆ηN,id ˙ξ + N ∆ξN,id ˙η
where
ξN,i △= ξ + N ∆vx− ξi, ηN,i △= η + N ∆vy− ηi. It is an easy matter to see thatdhi,d(hi◦ f ), and d(hi◦ f2) are mutually independent but, for anyN ≥ 3, one can write d(h◦fN) = β1N,i(x)dhi+β N,i 2 (x)d(hi◦f )+β N,i 3 (x)d(hi◦f2) where β1N,i(x) △ = ξ 0,i+ η0,i ξN,i+ ηN,i 1 +N 2− 3N 2 β2N,i(x) △ = ξ 1,i+ η1,i ξN,i+ ηN,i 2N − N 2 β3N,i(x) △ = ξ 2,i+ η2,i ξN,i+ ηN,i N2− N 2 .
Hence, the dimension of span(dΘ) is 3. As suggested in
[10], in general the differential dΘ can be a useful
start-ing point for constructstart-ing the observability decomposition. However, in this case, this is not necessary since the decom-position becomes apparent when the system is rewritten in terms of modified polar (MP) coordinates [9], [12], [13]. The MP state vector is defined as
qi= col ˙θi, ˙ri ri, θ i, 1 ri . (18)
whereθidenotes the bearing-angle measured from theξ-axis counterclockwise, ˙θi is the angular velocity,ri is the range from sensor position, and ˙ri the range-rate. Figure 1 shows the target-sensor geometry for the MP coordinates system.
η ξ ˙ri ˙θi ri θi
Fig. 1. MP Coordinate System sensor-target geometry
The Cartesian state vectorxk and the MP state vectorqik are related at all times by the following nonlinear one-to-one transformation qi= (ξ − ξi) ˙η − (η − ηi) ˙ξ (ξ − ξi)2+ (η − ηi)2 (ξ − ξi) ˙ξ + (η − ηi) ˙η (ξ − ξi)2+ (η − ηi)2 atan2 η − ηi, ξ − ξi 1/p(ξ − ξi)2+ (η − ηi)2 . (19)
It can be shown that only the first three components of the vectorqi are observable from a single non-moving DOA sensor [13]. In fact, by definingqi,o △= col( ˙θi, ˙ri/ri, θi) and
qi,¯o △= 1/ri, the dynamics of qi,o can be written as in (14) and, of course, the measurement equations take the form
yki = Cqi,o+ vki.
where C △= [0 0 1]. Further, in this particular case, the
function g(·) in (14) takes the form gi(qi,¯o, qi,¯ow
k). Thus, taking into account the fact that all bearing sensors have a limited and known range for detecting the presence of an object, the quantity qi,¯ow
k can be treated as a disturbance with unknown but bounded covariance.
Notice that, notwithstanding the lack of observability from a single sensor, the target state can be reconstructed at the fusion node provided that the number of sensors s
that transmit their measurements is greater than or equal 2.
Actually two sensors may not be sufficient for observability if target and sensors are aligned.
III. DATA-DRIVEN STRATEGIES FOR ESTIMATE TRANSMISSION
Let us now consider the case of estimate transmission, supposing that each sensor i transmits, instead of its local
measurement yi
k, a local estimate. Taking into account the development of Section II-B, only the component of the state that is locally observable is estimated and sent to the fusion node. This means that each sensor computes an estimateqˆk|ki,o
nonlinear filtering recursion ˆ qi,ok|k, Pi k|k = upd(ˆqk|k−1i,o , Pi k|k−1, yki, Ri, ˜hi) ˆ qi,ok+1|k, Pi k+1|k = pred(ˆqi,ok|k, Pi k|k, Q, ˜f o,i, gi) (20)
Comparing such a recursion with (20), one can see that here the true measurement yi
k is considered instead of the virtual measurementzi
k. This means that the estimateqˆ i,o k|kis computed using all the locally available information. Then, the local estimate is transmitted according to a data-driven strategy of the form
cik= ( 0 , ifkˆqk|ki,o − ˜qi,ok k2 Wi k ≤ δ i, 1 , otherwise, (21)
where the vectors q˜ki,o, k ∈ Z+, the positive definite weight matrices Wi
k, k ∈ Z+, and the positive reals δi have to be chosen so as to ensure that the transmission rate constraint (3) is satisfied.
Also in this case, the general principles discussed at the beginning of Section II-B can be used as a guideline for choosing the vectors q˜ki,o, k ∈ Z+ and the positive definite weight matricesWki, k ∈ Z+. Accordingly, the idea is to set
˜
qi,ok equal to the best prediction ofqki,othat can be computed on the basis of the information that is common to both sensor
i and the fusion node up to time k − 1. In particular, if one
denotes byni
k≥ 0 the number of time instants elapsed from the last transmission of sensori, i.e., ni
kis such thatcik−ni k =
1 and ci
k−1 = · · · = cik−ni k+1
= 0, then a natural choice
corresponds to settingq˜ki,o= ˆqk|k−ni,o i k
. Such a prediction can be recursively computed as ˆ qi,o k|k−ni k , Pk|k−ni i k = ( pred(ˆqi,o k−1|k−ni k−1 , Pk−1|k−ni i k −1 , Q, ˜fo,i, gi) if cik−1= 0 ˆ qi,ok|k−1, Pki|k−1 if c i k−1= 1
As for the weight matrixWi
k, one can compute, for example through the unscented transform, an approximation, sayΛi
k, of the covariance ofqˆi,ok|k− ˆqk|k−ni,o i
k
and then set
(Wi k)−1= 1 tr(Λi k) Λi k
where, as in (8), the normalization is a means for ensuring that the volume of the non-transmission region remains uniformly bounded.
As for the derivation of the fused estimate xˆk|k in the fusion node, we adopt an approach based on the idea of interpreting each local estimateqˆk|ki,o as a measurementzi
k of the true statexkcollected through the (virtual) measurement channel
zki = Ti,o(xk) + ǫik (22) where, the estimation errorǫi
k △
= ˆqk|ki,o − Ti,o(x
k) plays the role of a (virtual) measurement noise with covariance equal to the estimation error covariancePi
k|k. While, equation (22) can be directly applied only for the indices i for which an
estimate has been received (i.e., for whichci
k = 1), one can once again exploit the fact that, for a data-driven transmission
strategy, even in case of no transmission it is known that the estimateˆxi
k|k belongs to a certain ellipsoidδiEWk. Thus, ne
can treat the case of no transmission by replacing (22) with
zki = Ti,o(xk) + ǫik− uik (23) where zi
k △
= ˜qki,o and ui
k is uniformly distributed in the ellipsoid δiE
Wk. Summing up, by defining
zi k △ = ci kqˆk|ki,o + (1 − c i k)˜qi,ok , i = 1, . . . , s , Rik △ = Pk|ki + (1 − cik) δi ni,o+ 2(W i k)−1, i = 1, . . . , s , zk △ = col zk1, . . . , zsk , Rk △ = diag R1k, . . . , Rks , h(·) =△ col T1,o(·), . . . , Ts,o(·) ,
the estimate xˆk|k at the fusion node can be computed by means of a recursion which is formally analogous to (7).
Finally, observe that, because of the common process noise, the estimates qˆi,ok|k are actually dependent. The naive independence assumption has been adopted since standard techniques for fusing correlated information (e.g., covariance intersection) cannot be applied to the nonlinear and non-fully observable case. The development of alternative fusion rules (possibly based on the moving horizon estimation paradigm [14]) will be subject of future research.
IV. SIMULATION RESULTS
The goal of this section is to evaluate the performance of the proposed transmission strategies when applied to the network of bearing sensors described in section II-C. The covariance matrix of process noise has been assumed equal toQ = ζQ0 with Q0= ∆3/3 0 ∆2/2 0 0 ∆3/3 0 ∆2/2 ∆2/2 ∆ 0 0 0 0 ∆2/2 ∆ ,
ζ > 0 and sampling interval ∆ = 0.1 s. The following
transmission strategies have been compared:
• Data-Driven Measurement Transmission (DDMT):
the strategy of Section II; the thresholdsδi have been tuned so as to obtain the desired transmission ratesαi. • Periodic Measurement Transmission (PMT): sensors transmit periodically their measurements once every
1/αi time instants, the phase shift among the sensors being random.
• Data-Driven Estimate Transmission (DDET): the
strategy of Section III; the thresholdsδihave been tuned so as to obtain the desired transmission ratesαi. • Periodic Estimate Transmission (PET): sensors
trans-mit periodically their estimates once every 1/αi time instants, the phase shift among the sensors being ran-dom.
The comparison among the four strategies has been carried out via Monte Carlo simulations with independent runs
obtained for random generated trajectories by varying the measurement and process noises realizations. The time
aver-aged square error (TASE) at the fusion node F , T ASE =
1 T
PT
k=1(xk− ˆxk|k)′(xk− ˆxk|k), has been computed for each simulation run and for each strategy. Then, the mean and the maximum of the TASE over the Monte Carlo runs have been considered as performance indices. The simulation time
T and the number of Monte Carlo runs have been chosen
equal to600 and 1000, respectively.
Consider a network with eight sensors located at
(−16.67 + 16.67i, 0) for i = 0, 1, . . . , 7. A target moves in
the first quadrant and the target’s trajectory is generated so as to guarantee observability at each time instant from the overall sensor network. In all simulations the transmission rateαi has been fixed to0.1 for all sensors and transmission strategies. The target initial state is [60, 60, 0.1, 0.1]′, the variance of the measurement noise R = 3 10−4 rad (1◦ standard deviation), the variance of the process noise ζ = 10−3. The simulation results are shown in Table I. It can be noticed that transmitting the estimates is better than transmitting the measurements and data-driven strategies are better than the periodic ones. These results are in agreement with those obtained in [5] by the same strategies in the linear case. The relative percentage of TASE reduction of data-driven w.r.t. periodic strategies for both cases, transmitting the estimates and transmitting the measurements, have also been reported in Table I to help in the comparison.
A difference between the linear case treated in [5] and bearing-only tracking is that the transmission rate depends on the target trajectory. In particular, we have noticed that for the data-driven strategies (especially for DDET) the transmission rate increases at the decrease of the distance between target and sensor. In other words, sensors transmit more frequently when the distance target-sensor decreases. This means that data-driven strategies perform a natural sensor-scheduling in the nonlinear case. Sensors that are closer to the target (and, thus, sensors that have a more accurate estimate of the state) transmit more frequently. This is an interesting feature of data-driven strategies that we intend to investigate in future work. Notice that, for a fair comparison between the strategies with a constant transmission rate (the periodic ones) and the data-driven strategies, all simulations have been performed by generating trajectories that ensure that the average transmission rate αi is equal to the fixed value
0.1 for all sensors and transmission strategies (i.e., the target
is always “far enough” from the sensors). V. CONCLUSIONS
The paper has provided a contribution towards energy-efficient management of a centralized wireless sensor net-work employed for estimating the state of a dynamical system. An interesting solution to this problem has been provided in [5] but this solution is not applicable in certain situations of practical interest, like e.g. angle-only or rangle-only or Doppler-rangle-only target tracking, wherein there are nonlinear sensors that cannot fully observe the system. This
TASE PMT DMT PET DDET
s= 2 avg 0.8532 0.2660 (68%) 0.4483 0.1997 (55%) max 3.6511 1.0250 (71%) 3.3815 0.7300 (79%) s= 4 avg 0.5864 0.1819 (68%) 0.4213 0.1500 (63%) max 2.6720 0.6188 (76%) 2.9492 0.6238 (78%) s= 6 avg 0.4367 0.1364 (68%) 0.4081 0.1260 (69%) max 1.5589 0.5010 (67%) 3.1153 0.6301 (78%) s= 8 avg 0.3825 0.1065 (72%) 0.3633 0.1011 (72%) max 1.3968 0.3510 (74%) 2.4125 0.4182 (82%) TABLE I
CASE2:MEDIUM PROCESS NOISE VARIANCE
paper has filled this practical gap, providing novel data-driven transmission strategies that can be used for general collectively observable (i.e. observable from the whole set of sensors but non necessarily from a single one) non-linear systems. Future work along this line will concern: (1) centralized fusion techniques which do not rely on the independence assumption among local estimates, and (2) nonlinear observability decompositions for other types of sensors (e.g., Doppler-only).
REFERENCES
[1] J. P. Hespanha, P. Naghshtabrizi, and Y. Xu, “A survey of recent results in networked control systems,” Proc. of IEEE, Special Issue
on Technology of Networked Control Systems, vol. 95, pp. 138–162,
2007.
[2] Y. S. Suh, V. H. Nguyen, and Y. S. Ro, “Modified Kalman filter for networked monitoring systems employing a send-on-delta method,”
Automatica, vol. 43, pp. 332–338, 2007.
[3] Y. Mo, E. Garone, A. Casavola, and B. Sinopoli, “Stochastic sensor scheduling for energy constrained estimation in multi-hop wireless sensor networks,” IEEE Transactions on Automatic Control, vol. 56, pp. 2489 –2495, oct. 2011.
[4] G. Battistelli, A. Benavoli, and L. Chisci, “State estimation with remote sensors and intermittent transmissions,” Systems & Control
Letters, vol. 61, no. 1, pp. 155 – 164, 2012.
[5] G. Battistelli, A. Benavoli, and L. Chisci, “Data-driven communication for state estimation with sensor networks,” Automatica, vol. 48, no. 5, pp. 926–935, 2012.
[6] V. Gupta, T. H. Chung, B. Hassibi, and R. M. Murray, “On a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage,” Automatica, vol. 42, pp. 251–260, 2006.
[7] L. Zuo, R. Niu, and P. Varshney, “Posterior CRLB based sensor selection for target tracking in sensor networks,” in Acoustics, Speech
and Signal Processing, 2007. ICASSP 2007. IEEE International Con-ference on, vol. 2, pp. II–1041 –II–1044, april 2007.
[8] S. Julier, J. Uhlmann, and H. Durrant-Whyte, “A new method for the nonlinear transformation of means and covariances in filters and estimators,” IEEE Transactions on Automatic Control, vol. 45, pp. 477 –482, mar 2000.
[9] B. Ristic, S. Arulampalam, and N. Gordon, Beyond the Kalman filter:
particle filters for tracking applications. Artech House Publishers,
2004.
[10] F. Albertini and D. D’Alessandro, “Remarks on the observability of nonlinear discrete time systems,” in System Modeling and Optimization (J. D. et al., ed.), London: Chapman and Hall, 1996.
[11] U. Kotta and K. Schlacher, “Possible non-integrability of observable space for discrete-time nonlinear control systems,” in Proceedings of
the 17th IFAC World Congress, pp. 9852–9856, 2008.
[12] V. Aidala and S. Hammel, “Utilization of modified polar coordinates for bearings-only tracking,” IEEE Transactions on Automatic Control, vol. 28, pp. 283 – 294, mar 1983.
[13] A. Farina, “Target tracking with bearings only measurements,” Signal
Processing, vol. 78, no. 1, pp. 61 – 78, 1999.
[14] A. Alessandri, M. Baglietto, and G. Battistelli, “Moving-horizon state estimation for nonlinear discrete-time systems: new stability results and approximation schemes,” Automatica, vol. 44, no. 7, pp. 1753 – 1765, 2008.
